Introduction to Probability
Charles M. Grinstead
Swarthmore College
J. Laurie Snell
Dartmouth College
To our wives
and in memory of
Reese T. Prosser
Contents
1 Discrete Probability Distributions
1.1 Simulation of Discrete Probabilities . . . . . . . . . .
Chapter 6
Plane strain problems
6.1
Basic equations
Denition: A deformation is said to be one of plane strain (parallel to the plane x 3 = 0) if:
u3 = 0 and
u = u (x ).
(6.1)
There are only two independent variables, (x1 , x2 ) = (x, y).
Plane strain is
Stress distribution in plate with a circular hole (From:
Eschenauer, H. & Schnell. W. Elastizitaetstheorie I,
BI Wissenschaftsverlag, 1986)
<> ; 0 =T
Note that the hoop stress at the hole is three times larger
than the stress applied at infinity. In engi
Chapter 4
Elasticity & constitutive equations
4.1
The constitutive equations
The constitutive equations determine the stress ij in the body as function of the bodys deformation.
Denition: A solid body is called elastic if
ij (xn , t) = ij (ekl (xn , t).
eij
=
ij
=
1
2
1
2
uj
ui
+
xj
xi
ui
uj
xj
xi
3
= ji
= eji
is the strain tensor and
(2.5)
is the rotation tensor.
(2.6)
where
ui
= eij + ij ,
xj
ui
xj
(2.4)
is the displacement gradient tensor:
ui
xj
1.
(2.3)
We will restrict ourselves to a linearised an
7
2 ui
ij
+ Fi = 2 ,
xj
t
(3.4)
Including inertial eects via DAlembert forces gives the equations of motion:
ij
+ Fi = 0.
xj
(3.3)
The equations of equilibrium for a body, subject to a body force (force per unit volume) F i is
3.3
The equations of equi
1
MT30271: EXAMPLE SHEET1 IV
1.) A cylinder which occupies the region x2 + x2 R2 and L x3 0 is in static
1
2
equilibrium. The stress tensor in the body is given by 11 = 12 = 22 = 0 and
13 = ax2 , 23 = ax1 , 33 = bx3 , where a and b are known (small) const
1
MT30271: EXAMPLE SHEET1 I
1.) Which one of these equations in index notation are valid? Remember the summation
convention!
a) c = ai bi
b) c = aij bi
c) ci = aij bi
d) ci = aij bj
e) ci = aji bj
f ) ij = ij T + Eijkl ekl
g) ij = kl Ti + Eijkl eij
h) kij
1
MT30271: EXAMPLE SHEET1 V
1.) Show that in a homogeneous, isotropic, linearly elastic body the principal axes of the
stress and strain tensors coincide. Derive the relation between principal strains and
principal stresses.
2.) It was shown in the lectur
1
MT30271: EXAMPLE SHEET1 II
1.) A 2D body occupying the region cfw_d : 0 x1 1, 0 x2 1 is displaced by the
following displacement eld (see problem 3 on the last sheet) u1 = (x1 + 2x2 ); u2 =
(3 + x2 ) where
1.
a) Compute the extension en1 of a line elemen
1
MT30271: EXAMPLE SHEET1 III
1.) The strains eij (r) are given throughout a body. Show that the corresponding displacement eld ui (r) is only determined to within a rigid-body displacement. Use
the following two steps: (i) Show that if two displacement e
Essential Dierential Equations
Solutions 6
1. The generic solution of the ODE system du = Au is given by u(t) =
dt
eAt u(0). In this example we have the symmetric matrix
1 2
2 1
A=
.
The simplest way to solve the ODE system is to diagonalise A by computin
Essential Dierential Equations
Time 15.0515.55
Closed book test: 05112013
Attempt all six questions
No Calculators
(1) Consider a vector space X. Give the denition of a norm
If X = R2 , which of the following are norms?
x
1
=
1
(|x1 | + |x2 |),
2
x
2
= mi
Examples 6
Essential Dierential Equations
1. Write down the generic solution of the following ODE system: nd u(t)
R2 such that
du
=
dt
1 2
2 1
u;
t > 0;
with
u(0) =
1
2
.
Can you determine an explicit solution? (Hint: to do this you need to
calculate the
Solutions 5
Essential Dierential Equations
1. Note that u(j) (x) = sin(jx) satises the boundary condition u(j) (1) = 0
if and only if j is an integer. Moreover, if j = 0 then the solution is
identically zero and if j is positive then eigenfunctions u(j) c
Examples 5
Essential Dierential Equations
Consider the following eigenvalue problem: nd the pair cfw_, u, where R
and u : (0, 1) R, such that
u (x) = u(x)
x (0, 1),
(E)
together with the boundary conditions u(0) = 0 = u(1).
1. Show that there are innitely
Solutions 1
Essential Partial Dierential Equations
1. The six functions f1 , f2 , . . . , f6 are plotted below. The functions with
positive exponents are continuous over [0, 1]. The functions with negative
exponent are unbounded in the limit x 0 so are no
Examples 4
Essential Dierential Equations
1. Let V := cfw_v|v H 1 (0, 1); v(0) = 0, v(1) = 0 and consider the following
variational problem: given
f (x) =
1, 0 < x 1/2,
0, 1/2 < x < 1,
nd u V such that
a(u, ) = (),
where
V,
()
1
1
u
a(u, ) =
and
f.
()
Solutions 2
Essential Partial Dierential Equations
Ax
Ax
1. Let x = 0 be the function that satises x = supx=0 x . Note that
Ax / x A for any specic function x X. To show that A is a
valid norm we simply check the axioms:
positivity
A =
Ax
( 0)
=
0
x
(> 0
Solutions 4
Essential Dierential Equations
1
1. (a) We have u = 1 for x 1/2 and u(0) = 0. Hence, u(x) = 2 x2 +bx
= 0 for x > 1/2 and u(1) = 0 so that
for some b. Similarly, u
u(x) = a(x 1) for some constant a.
If we enforce the conditions that u and u ar
Solutions 3
Essential Dierential Equations
1. Given that Xh X we know that vh X so that
a(u, vh ) = (vh ) = a(uh , vh ) a(u, vh ) a(uh , vh ) = 0.
Since a(, ) is bilinear the result follows.
2. First, since a is coercive on X, we know that there exists >
Examples 1
Essential Partial Dierential Equations
1. Using MATLAB or otherwise, plot the following functions f : (0, 1) R
and list the ones that are continuous on [0, 1].
(i) f1 (x) = x2
(ii) f2 (x) = x3/2
(iii) f3 (x) = x1/2
(iv) f4 (x) = x1/4
(v) f5 (x)
3
THE CONVECTIONDIFFUSION EQUATION
We next consider the convectiondiusion equation
2
u+w
u = f,
(3.1)
where > 0. This equation arises in numerous models of ows and other physical
phenomena. The unknown function u may represent the concentration of a pollu
Finite Elements and Fast Iterative Solvers: With
Applications in Incompressible Fluid Dynamics
Howard C. Elman, David J. Silvester and
Andrew J. Wathen
2005
MIMS EPrint: 2007.156
Manchester Institute for Mathematical Sciences
School of Mathematics
The Uni
APPROXIMATION BY FINITE ELEMENTS
123
where the continuity constant is w = + w L. The continuity of (v) of
(3.15) may be established using the trace inequality of Lemma 1.5, and the
norm equivalence in Proposition 1.6,
| (v)| f
f
v +
v
1,
gN
N
v
+ C gN
C
Computational Exercises III
MATH46052|66052
Numerical solutions to a convectiondiusion problem dened on a square domain can be computed using the ifiss software package in two steps.
First, running square cd (or ref cd to give the option of a Neumann cond